Polar Coordinates, Polar Derivatives, and Areas in Polar Form

The Cartesian grid places every point by its signed distances $(x, y)$ from two perpendicular axes. The polar grid instead specifies a point by its radial distance $r$ from the origin and its angle $\theta$ from the positive $x$-axis. Many curves — spirals, rose curves, cardioids, and above all conic sections — have much simpler polar equations than Cartesian ones. In quantitative finance, polar coordinates appear behind the scenes in the derivation of the Gaussian integral $\int_{-\infty}^\infty e^{-x^2}\,dx = \sqrt{\pi}$ (via polar double integral), and the area-in-polar-form formula $A = \frac{1}{2}\int_\alpha^\beta r^2\,d\theta$ has the same Riemann-sum structure as a European option's payoff expressed as a sector average.

Learning objectives. By the end of this note you should be able to: